Equivalence relation
From Encyclopedia of Mathematics
A binary relation on a set
with the following properties:
1) for all :
(reflexivity);
2) (symmetry);
3) (transitivity).
If maps the set
into a set
, then the relation
is an equivalence.
For any the set
consisting of all
equivalent to
is called the equivalence class of
. Any two equivalence classes either are disjoint or coincide, that is, any equivalence defines a partition (decomposition) of
, and vice versa.
How to Cite This Entry:
Equivalence relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equivalence_relation&oldid=17622
Equivalence relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equivalence_relation&oldid=17622
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article